Computer aided engineering analysis, for example, topology optimization techniques such as finite element models (FEM), incorporate computational techniques that are used to find approximate solutions to engineering problems. In general, a model representing the geometry of interest is discretized into a plurality of elements that collectively represent the entire geometry. Because of the elements' reduced size, computers executing the FEM analysis are capable of solving the partial differential equations that govern the physical properties of each of the elements and, therefore, provide an approximate solution to the physical problem for the entire geometry of interest.
Defining in the FEM model the orientation of the material physical property or properties (e.g., magnetic north of a permanent magnet) in polar coordinates (i.e., radius and angle of orientation) imposes a non-continuous design constraint on the material physical property because the angle of orientation is between 0° and 360°. Similarly, limiting the orientation of the material physical property using a Cartesian coordinate system imposes a nonlinear design constraint on the property because the position of the property must satisfy a second-order condition (i.e., (x2+y2)1/2<1).
Another difficulty in optimizing the orientation of material physical properties includes application of design constraints that are non-continuous. For example, properties may be limited in orientation to a pre-determined number of directions in the physical domain. For example, manufacturing constraints may limit the orientation of the property, such as the direction of magnetic fields of magnets in a magnetic system, into four, six, or eight directions.
Using traditional optimization techniques, the material properties of each of the finite elements would have non-continuous design constraints applied, which increases optimization model complexity and computational time to complete convergence of the optimization model. Further, incorporating such a model with non-continuous design constraints into an optimization routine may be difficult or prohibitive, as the optimization model having non-continuous design constraints may lack the flexibility to make incremental changes to the orientation of the material physical properties of the elements while allowing the optimization model to converge.
Accordingly, a need exists for alternative methods for convergence of nonlinear programming problems that involve angles as material property design variables, and promoting convergence of the material property design variable angles into a set of specified desired angles.